CHEVALIER G. DE PAMBOUR'S
NEW THEORY
OF THE
STEAM ENGINE.
PRICE 2s.
LONDON :— JOHN WEALE. 1838.
A NEW THEORY
OF THE
STEAM ENGINE,
AND THE
MODE OF CALCULATION BY MEANS OF IT,
OF
THE EFFECTIVE POWER, &c. OF
EVERY KIND OF STEAM ENGINE, STATIONARY
OR LOCOMOT^V^ * V ,
BY
qiMt?
THE CHEVALIER G./t)E PAMBOUR.
LONDON:
JOHN WEALE, ARCHITECTURAL LIBRARY,
59, HIGH HOLBORN.
1838.
or the proportions of steam-engines ; and a succinct
exposition of the method proposed.
Part ii. — General formulae for the calculation of the
effects, &c. of rotative, stationary or locomotive, high
or low pressure, expansive or unexpansive, condensing
or uncondensing steam-engines, according to the pro-
posed theory.
Part in.— Special application of these formulae to
the divers systems of steam-engines in use.
PART I.
PROOFS OF THE INEXACTITUDE OF THE ORDINARY ME-
THODS, AND EXPOSITION OF THE ONE PROPOSED.
§ 1. Mode of calculation hitherto in use. — All the
problems in the application of steam-engines merge
into these three —
The velocity of the motion being given, to find the
load the engine will move at that velocity.
The load being given, to find the velocity at which
the engine will move that load ;
And, the load and the velocity being given, to find
the vaporization necessary, and consequently the
area of heating surface requisite for the boiler, in
order that the given load be set in motion at the
given velocity.
The problem, which consists in determining the
useful effect to be expected from an engine of which
the number of strokes of the piston per minute is
counted, that is, whose velocity is known, evidently
amounts to determining the effective load correspond-
ing to that velocity ; for that load being once known,
by multiplying it by the velocity we have the useful
effect required.
According to the mode of calculation hitherto ad-
mitted, when it is wanted to know the useful effect an
engine will produce at a given velocity, or, in other
words, the effective load that it will set in motion at
that velocity, the area of the cylinder is multiplied by
the velocity of the piston, and that product by the
pressure of steam in the boiler ; this gives, in the first
place, what is called the theoretical effect of the
engine. Then, as experience has shown that steam-
engines can never completely produce this theoretical
effect, it is reduced in a certain proportion, indicated
by a constant number, which is the result of a com-
parison between the theoretical and practical effects of
some engines previously put to trial ; and thus is
obtained the number which is regarded as the prac-
tical effect of the engine, or the work it really ought
to execute.
A mode perfectly similar is followed, for deter-
mining the vaporization which an engine ought to
produce in order to produce a desired effect ; that is
to say, for resolving the third of the problems which
we have presented above. As to the second of these
problems, that which consists in determining the
velocity the engine will assume under a given load, no
solution of it has been proposed in this way, and we
shall expose, farther on, some fruitless essays that
have been made to resolve it in another way.
As in the above-mentioned calculation no account is
taken of friction, nor of some other circumstances
which appear likely to diminish the power of the
engine, the difference observed between the theoretical
and the practical result excites no surprise, and is
readily attributed to the circumstances neglected in the
calculation.
§ 2. First objection against this method of calculation.
— This mode of calculation is liable to many ob-
jections, but for the sake of brevity we limit ourselves
to the following : —
The coefficient adopted to represent the ratio of the
practical effects to the theoretical, varies from ^ to -f ,
according to the various systems of steam-engines ;
that is to say, that from f to ^ of the power exerted
by the machine is considered to be absorbed by friction
and divers losses. Not that this friction and these
losses have been measured and found to be so much,
but merely because the calculation that had been
made, and which might have been inexact in prin-
ciple, wanted so much of coinciding with experience.
Now it is easy to demonstrate, that the friction and
losses which take place in a steam-engine can never
amount to f , nor to •§• of the total force it developes.
It will suffice to cast an eye on the explanation at-
tempted, on this point, by Tredgold, who follows this
method in his Treatise on S team-Engines.* He says
(art. 367), that, for high pressure engines, a deduction
* The author here refers to the first edition of ' Tredgold on the
Steam-Engine :' in the new edition just published the algebraic
parts are transformed by the editor into easy practical rules, accom-
panied by examples familiarly explained for the working engineer.
of H) must be made from the total pressure of the
steam, which amounts to a deduction of f-0 on the
ordinary effective pressure of such engines ; and to
justify this deduction, which however is still not
enough to harmonize the theoretical and practical
results in many circumstances, he is obliged to esti-
mate the friction of the piston, with the losses or
waste, at ^ of the power, and the force requisite for
opening the valves and overcoming the friction of the
parts of the machine, at TO^ of that power. Reflecting
that these numbers express fractions of the gross power
of .the engine, we must readily be convinced that they
cannot be correct ; for, in supposing the engine had a
useful effect of 100 horses, which, from the reduction
or coefficient employed, supposes a gross effect of 200
horses, 12 would be necessary to move the machinery,
40 to draw the piston, &c. ! The exaggeration is
evident.
Besides, in applying this evaluation of the friction
to a locomotive engine, which is also a high pressure
steam-engine, and supposing it to have 2 cylinders of
12 inches diameter, and to work at 75 ft>s. total press-
ure, which amounts to 60 Ibs. effective pressure, per
square inch, we find that from the preceding estimate,
the force necessary to draw the piston would be 5650
Ibs., whereas our own experiments on tjie locomotive en-
gine, the Atlas, which is of these dimensions, and works
at that pressure, demonstrate that the force necessary
to move, not only the two pistons, but all the rest of
the machinery, including the waste, &c., is but 48 Ibs.
applied to the wheel, or 283 Ibs. applied on the piston.
8
It is then impossible to admit, that in steam-engines
the friction and losses can absorb the half, nor the
third, much less the f of the total power developed ;
and yet there do occur cases wherein, to reconcile the
practical effects with the theoretical ones thus calcu-
lated, it would be necessary to reduce the latter to the
fourth part, and even to less ; and what is more, it
often happens, that the same engine which in one case
requires a reduction of f , will not in other cases need
a reduction of more than about ^. This is observed
in calculating the effects of locomotive engines at very
great velocities, and afterwards at very small ones.
There is no doubt, -then, that the difference ob-
served between the theoretical effect of an engine
and the work which it really performs, does not arise
from so considerable a part of the applied force being
absorbed by friction and losses, but rather from the
error of calculating in this manner the theoretical
effect of the machine. In effect, this calculation sup-
poses that the motive force, that is, the pressure of
the steam against the piston or in the cylinder, is the
same as the pressure of the steam in the boiler ;
whereas we shall presently see, that the pressure in
the cylinder may be sometimes equal to that of the
boiler, sometimes not the half nor even the third of it,
and that it depends on the resistance overcome by the
engine.
§ 3. Formula proposed by divers authors to deter ~
mine the velocity of the piston under a given load, and
proofs of their inexactitude. — We have said that this
problem was not resolved by the foregoing method.
The following are the attempts made to that end by
another way. Tredgold, in his Treatise on Steam-
Engines (art. 127 and following), undertakes to cal-
culate the velocity of the piston from considerations
deduced from the velocity of the flowing of a gas,
supposed under a pressure equal to that of the boiler,
into a gas supposed at the pressure of the resistance.
He concludes from thence, that the velocity of the
piston would be expressed by this formula,
V = 6-5 V^-
in which V is the velocity in feet per second, and h
stands for the difference between the heights of two
homogeneous columns of vapour, one representing the
pressure in the boiler, the other that of the resistance.
But it is easily seen, that this calculation supposes the
boiler filled with an inexhaustible quantity of vapour,
since the effluent gas is supposed to rush into the
other with all the velocity it is susceptible of acquir-
ing, in consequence of the difference of pressure.
Now such an effect cannot be produced, unless the
boiler be capable of supplying the expenditure, how-
ever enormous it might be. This amounts conse-
quently to supposing that the production of steam in
the boiler is unlimited. But, in reality, this is far
from being the case. It is evident that the velocity of
the piston will soon be limited by the quantity of
steam producible by the boiler in a minute. If that
production suffice to fill the cylinder 200 times in a
minute, there will be 200 strokes of the piston per
minute ; if it suffice to fill it 300 times, there will be
300 strokes. It is then the vaporization of the boiler
10
which must regulate the velocity, and no calculation
which shall exclude that element can possibly lead to
the true result ; consequently the preceding formula
cannot be exact.
This is why, in applying this formula to the case
of an ordinary locomotive engine of the Liverpool
Railway with a train of 100 tons, the velocity the
engine ought to assume is found to be 734 feet per
second, instead of twenty miles an hour, or five feet
per second, which is its real velocity.
Again, in his Treatise on Railways (page 83), Tred-
gold proposes the following formula, without in any
way founding it on reasoning or on fact :
V = 240 VHP
W
in which V is the velocity of the piston in feet per
minute, I the stroke of the piston, P the effective
pressure of the steam in the boiler, and W the resist-
ance of the load. But as this formula makes no
mention either of the diameter of the cylinder, or
of the quantity of steam supplied by the boiler in a
minute, it clearly cannot give the velocity sought ; for
if it could, the velocity of an engine would be the
same with a cylinder of one foot diameter as with a
cylinder of four feet, which expends sixteen times as
much steam. The area of heating surface, or the
vaporization of the boiler, would be equally indif-
ferent : an engine would not move quicker with a
boiler vaporizing a cubic foot of water per minute,
than with one that should vaporize but ^ or aV
Hence this formula is without basis.
11
Wood, in his Treatise on Railways (page 351), pro-
poses the following formula also, without discussion,
m
where V is the velocity of the piston in feet per minute,
I the length of stroke of the piston, W the resistance
of the load, and P the surplus of the pressure in the
boiler, over and above what is necessary to balance
the load W. This formula being liable to the same
objections as the preceding, is also demonstrated in-
admissible a priori.
Consequently, of the three fundamental problems
of the calculation of steam-engines, two have received
inaccurate solutions by means of the coefficients, and
the third, as we have just seen, has received no solu-
tion at all.
§ 4. Succinct exposition of the proposed theory. —
After having made known the present state of science,
with regard to the theory and estimation of the effect-
ive power of steam-engines, it remains to exhibit the
theory we apply to them ourselves.
It is well known, that in every machine, when the
effort of the motive power becomes superior to the
resistance, a slow motion is created, which quickens
by degrees till the machine has attained a certain
velocity, beyond which it does not go, the motive
power being incapable of producing greater velocity
with the mass it has to move. Once this point at-
tained, which requires but a very short space of time,
the velocity continues the same, and the motion
remains uniform as long as the effort lasts. It is from
12
this point only that the effects of engines begin to
be reckoned, because they are never employed but
in that state of uniform motion; and it is with
reason that the few minutes, during which the velo-
city regulates itself, and the transitory effects which
take place before the uniform velocity is acquired, are
neglected.
Now, in an engine arrived at uniform motion, the
force applied by the motive power forms strictly an
equilibrium with the resistance ; for if that force were
greater or less, the motion would be accelerated or
retarded, which is contrary to the hypothesis. In a
steam-engine the force applied by the motive agent
is nothing more than the pressure of the steam against
the piston, or in the cylinder. The pressure therefore
in the cylinder is strictly equal to the resistance of the
load against the piston.
Consequently the steam, in passing from the boiler
to the cylinder, may change its pressure, and assume
that which is represented by the resistance of the
piston. This fact alone exposes all the theory of the
steam-engine, and in a manner lays its play open.
From what has been said, the force applied on the
piston, or the pressure of the steam in the cylinder,
is therefore strictly regulated by the resistance of the
load against the piston. Consequently calling F the
pressure of the steam in the cylinder, and R the
resistance of the load against the piston, we have as a
first analogy,
P' = R.
To obtain a second relation between the data and
13
the qusesita of the problem, we shall observe that
there is a necessary equality between the quantity of
steam produced, and the quantity expended by the
machine ; the proposition is self-evident. Now if
we express by S the volume of water vaporized in the
boiler per minute, and effectively transmitted to the
cylinder, and by ra the ratio of the volume of the
steam generated under the pressure P of the boiler,
to the volume of water which produced it, it is clear
that
• mS
will be the volume of steam formed per minute in the
boiler. This steam passes into the cylinder, and there
assumes the pressure P' ; but if we suppose that, in
this motion, the steam preserves its temperature in
passing from the boiler to the cylinder, or from the
pressure P to the pressure P', its volume increases in
the inverse ratio of the pressures. Thus the volume
m S of steam furnished per minute by the boiler will,
when transmitted to the cylinder, become
.,. ^ ^ : -a..*. -. • •
On another hand, v being the velocity of the piston,
and a the area of the cylinder, a v will be the volume
of steam expended by the cylinder in a minute.
Wherefore, by reason of the equality which necessarily
exists between the production of the steam and the
expenditure, we shall have the analogy of
av=m S.— ;
which is the second relation sought.
14
Consequently, by exterminating P' from the two
equations, we shall have as a definitive analytic rela-
tion among the different data of the problem :
~~a~ IT
This relation is very simple, and suffices for the
solution of all questions regarding the determination
of the effects or the proportions of steam-engines.
As we shall develope its terms hereafter, in taking
it up in a more general manner, we content ourselves
to leave it for the present under this form, which will
render the discussion of it easier and clearer.
The preceding equation gives us the velocity as-
sumed by the piston of an engine under a given
resistance R. If, on the contrary, the velocity of
the motion be known, and it be required to calculate
what resistance the engine will move at that velocity,
it will suffice to resolve the same equation with refe-
rence to R, which will give
mSP
Jx = .
a v
Finally, supposing the velocity and the load to be
given beforehand, and that it be desired to know what
vaporization the boiler should have to set the given
load in motion at the prescribed velocity, it will still
suffice to draw from that analogy the value of S,
which will be
g _ a v R
On these three determinations we rest for the moment,
because, as will soon appear, they form the basis of
15
all the problems that can be proposed on steam-
engines.
§ 5. New proofs of the exactitude of this theory,
and of the inaccuracy of tht ordinary mode of calcula-
tion.— The theory just developed demonstrates that
the steam may be generated in the boiler at a certain
pressure P, but that in passing to the cylinder it
necessarily assumes the pressure R, strictly deter-
mined by the resistance to the piston, whatever the
pressure in the boiler may be. Consequently, accord-
ing to the intensity of that resistance, the pressure in
the cylinder, far from being equal to that in the boiler,
or from differing from it in a certain constant ratio,
may at times be equal to it, and at other times very
considerably different. Hence those who, in perform-
ing the ordinary calculation, consider the force applied
on the piston as indicated by the pressure in the
boiler, begin by introducing into their calculation an
error altogether independent of the real losses to
which the engine is liable. To this cause, then, and
not to the friction and losses, which can form but
the smallest part of it, must be attributed the enor-
mous difference which, in this mode of calculation, is
found between the theoretical effect of the engine, and
the work which it really executes.
We have already proved the mode of action of the
steam in the cylinder by the consideration of uniform
motion ; but in examining what passes in the engine,
we shall immediately find many other proofs.
1st. The steam, in effect, being produced at a
16
certain degree of pressure in the boiler, passes into
the tube of communication, and thence into the
cylinder. It first dilates, because the area of the
cylinder is from ten to twenty-five times that of the
tube ; but it would promptly rise to the same degree as
in the boiler, were the piston immoveable. But as
the piston, on the contrary, opposes only a certain
resistance, determined by the load sustained by the
engine, it will yield as soon as the elastic force of the
steam in the cylinder shall have attained that point.
The piston, in consequence, will be a valve to the
cylinder. Hence the pressure in the cylinder can
never exceed the resistance of the piston, for that
would be supposing a vessel full of steam, in which
the pressure of the steam would be greater than that
of the safety valve.
2nd. Were it true, that the steam flowed into the
cylinder, either at the pressure of the boiler, or at
any other pressure which were to that of the boiler
in any fixed ratio, as the quantity of steam generated
per minute in the boiler would then flow at an iden-
tical pressure in all cases, and would consequently
fill the cylinder an identical number of times per
minute ; it would follow, that as long as the engine
should work with the same pressure in the boiler, it
would assume the same velocity with all loads. Now
we know that precisely the contrary takes place, the
velocity increasing when the load diminishes ; and
the reason of it is, that when the load is half, the
steam flowing also at a half pressure into the cylinder,
17
and consequently acquiring a volume double what it
had before, will serve for double the number of strokes
of the piston.
3rd. Applying the same reasoning inversely, we per-
ceive that were the pressure in the cylinder really
bearing a constant ratio to that in the boiler, or if it
be preferred, constant so long as that in the boiler did
not vary, we should, in calculating the effort of which
the engine would be capable, always find it the same,
whatever be the velocity of the piston. Thus, at any
velocity whatever, the engine would always be capable
of drawing the same load ; which experience again
contradicts, for the greater the velocity of the piston,
the lower the pressure of the steam in the cylinder,
whence results, that the load of the engine lessens at
the same time.
4th. Another no less evident proof of this is easily
adduced. Were it true that the pressure in the
cylinder were to that in the boiler in any fixed pro-
portion, since the same locomotive engine always
requires the same number of revolutions of the wheel,
or the same number of strokes of the piston to tra-
verse the same distance, it would follow that, as long
as those engines worked at the same pressure, they
would consume iri all cases the same quantity of
water for the same distance. Now the quantity of
water, far from remaining constant, decreases on the
contrary with the load, as may be seen by the experi-
ments we have published on this subject. Here
therefore again it is proved, that, notwithstanding the
equality of pressure in the boiler, the density of the
18
steam expended follows the intensity of the resistance,
that is to say, the pressure in the cylinder is regulated
by that resistance.
5th. Similarly, the consumption of fuel being in
proportion to the vaporization effected, it would fol-
low, if the ordinary theory were exact, that the quan-
tity of fuel consumed by a given locomotive, for the
same distance, would always be the same, with what-
ever load. Now we again find by experience that
the quantity of fuel diminishes with the load, con-
formably to the explanation we have given of the
effects of the steam in the engine.
6th. It is again clear that if the pressure in the
cylinder were, as it is believed, constant for a given
pressure in the boiler, that so soon as it was recog-
nised that an engine could draw a certain load with
a certain pressure, and communicate to it a uniform
motion, it would follow, that the same engine could
never draw a less load with the same pressure, with-
out communicating to it a velocity indefinitely ac-
celerated ; since the power, having been found equal
to the resistance of the first load, would necessarily
be superior to that of the second. Now experience
proves, that in the second case the velocity is greater,
but that the motion is no less uniform than in the
first ; and the reason of this is, that though the steam
may indeed be produced in the boiler at a greater or
less pressure, and that it matters little, yet on passing
into the cylinder, it always assumes the pressure of
the resistance, whence results that the motion must
remain uniform as before.
19
7th. Finally, in looking over our experiments on
locomotives, it will be seen that the same engine will
sometimes draw a light load with a very high pressure
in the boiler, and sometimes a heavy load with a very
low pressure. It is then impossible to admit, as the
ordinary calculation supposes, that any fixed ratio
whatever has existed between the two pressures.
Moreover, the effect just cited is easy to explain, for
it depends simply on this, that in both cases the
pressure in the boiler was superior to the resistance on
the piston ; and it needed no more for the steam,
generated at that pressure or at any other, satisfying
merely that condition, to pass into the cylinder and
assume the pressure of the resistance.
It is then visible, from these various proofs, that
the pressure in the cylinder is strictly regulated by
the resistance on the piston, and by nothing else ;
and that any method like that of the coefficients in
the ordinary calculation, which tends to establish a
fixed ratio between the pressure in the cylinder and
that of the boiler, must necessarily be inexact.
$ 6. Verification of the two modes of calculation
by particular examples. — We have sufficiently demon-
strated the want of basis of the ordinary calculation ;
but as the inaccuracy we have just exposed in that
method might by some be supposed to be of slight
importance, and they might conceive that, in practical
examples, it amounted to the obtaining of results,
which if not quite exact, were at least very near the
truth, we will now attempt to apply it to some par-
ticular cases.
20
The coefficient of reduction for high pressure en-
gines, working without expansion and without con-
densation, not being given by the authors who have
treated on these subjects, we propose, in order to
determine it, the two following facts, which took place
before our eyes : —
i. The Leeds locomotive engine, which has two
cylinders eleven inches in diameter, stroke of the
piston sixteen inches, wheel five feet in diameter, drew
a load of 88 '34 tons, in ascending a plane inclined
1 in 1300, at the velocity of 20'34 miles an hour;
the effective pressure in the boiler being 541fos. per
square inch, or the total pressure 68*71 ibs. per square
inch.
ii. The same day, the same engine drew a load of
38' 52 tons in descending a plane inclined 1 in 1094, at
the velocity of 29 '09 ; the pressure in the boiler being
precisely the same as in the preceding trial, and the
regulator open to the same degree. These experiments
may be seen in pages 233 and 234 of our Treatise on
Locomotives.
If on one hand be reckoned, according to the
ordinary method, the theoretic effort applied to the
piston, and on the other hand the effect really pro-
duced, viz., the resistance opposed by the load plus
that of the air against the train, we find, on refer-
ring the pressure and the area of the pistons to the
foot square : —
21
1st case. — Theoretic effort applied on the
piston, according to the ordinary
calculation 1*32 x (6871 x 144) . 13,060 fts.
Real effect .' . * . . . . . 8,846
Coefficient of correction 0'68
2nd case. — Theoretic effort, the same as
above 13,060
Real effect . ... 1 *. ' . . 6,473
Coefficient of correction . . 0*50
The mean coefficient, to apply to the total pressure,
to convert the theoretic effects to the practical, is
then -59.
We find, then, three very different coefficients :
choose the first case, then an error occurs in the
second ; choose the second, and an error must arise in
the first ; by taking the third, you will only divide
the error between the two. In every way an error is
inevitable, and that alone suffices to prove that every
method, like the ordinary one, which consists in the
use of a constant coefficient, is necessarily inexact,
whatever be the coefficient chosen, and to whatever
engine the application be made ; for it is evident that
the same fact would occur in every kind of steam-
engine. Only that it might be less marked, if the
velocities at which the engine were taken were less
different ; and this is what has hitherto prevented the
error of this method from being perceived, for all the
engines of the same system being imitated from each
22
other, and moving nearly at the same velocity, the
same coefficient of correction seems tolerably to suit
them, from the factitious limit that had been laid down
for the speed of the piston.
Besides, in stationary engines, one cannot, for want
of precise determinations of the friction, disengage in
the result the part which is really attributable to it
from that which constitutes a positive error. But
here we may easily be convinced that neither of these
coefficients of correction represents, as the ordinary
theory would have it, the friction, losses, and various
resistances of the machine ; for direct experiments
made on the engine under consideration, and noted in
our Treatise on Locomotives, enable us to estimate
separately all these frictions, losses, and resistances.
Reckoning, then, the friction of the engine at 82 fts.,
taking account besides of its additional friction per ton
of load, and adding for each case the pressure subsist-
ing on the opposite side of the piston by the effect of
the blast pipe, we find, as the sum of the friction and
indirect resistances —
1st case.- — Friction 1,257 ft>s.
or • 10 of the theoretic result.
2nd case. — Friction 873 fos.
or '07 of the theoretic result.
Thus we see that in each of the two cases, the fric-
tion and indirect resistances, omitted in the calcula-
tion, do not in reality amount to more than 10 or 7
hundredths of the theoretic result ; and if we should
be disposed to add to that ^ or '05, for the filling of
23
the vacant spaces of the cylinder, which we could not
estimate in ffis., it will be *15 and *12; whereas the
coefficients of correction would raise them to *32 in
one case, and "50 in the other; that is, to 2 and 4
times what they really are. If, then, from these co-
efficients, be deducted the true value of the friction
and losses, it will appear that the theoretic error,
introduced into the calculation under the denomination
of friction, is 17 per cent, of the total power of the
engine in the one case, and 38 per cent, in the other.
But it is to be remarked, that, from the preceding
evaluations, viz., of the direct resistances first, and then
of the friction and indirect resistances, we have, for
each of the two cases in question, the sum of the total
effects really produced by the machine, as follows :—
1st case. — Direct resistances .... 8,846 Ibs.
Friction . .... . . 1,257
10,103
2nd case. — Direct resistances . . . . 5,473
Friction 873
6,346
We are therefore enabled now to compare these
effects produced with the results either of the ordinary
calculation or of our theory.
1°. In applying the ordinary calculation with the
mean coefficient ' 56 determined above, and comparing
its result with the real effect, we find —
24
1st case. — Effort applied on the piston,
according to the ordinary calcula-
tion, 1-32 x (6871 x 144) x '59 . 7,705 fos.
Effect produced, including friction
and every resistance . . • V. . 1 0, 1 03
Error over and above the friction and
resistances 2,398
2nd case. — Effort applied on the piston,
according to the ordinary calcula-
tion, the same as above . . . 7,705 tbs.
Effect produced, including friction
and every resistance - .... 7,346
Error over and above the friction and
resistances 359
Mean error of the two cases . . . 1,378
It is then evident what error would have been com-
mitted in calculating the effects of this engine from
the coefficient * 59 ; but it is equally evident, that in
applying any other coefficient whatever, the error would
onl]£ transfer itself from one case to the other, without
ever disappearing ; and thus it is that the coefficient
•59 has almost annulled the error of the second case,
by transferring it to the first.
To apply our formula with reference to the same
problem, viz. : —
a R = mS P
a v
we have nothing more to do than to substitute for the
letters their value, taking care to refer all the measures
to the same unit. In making then these substitutions,
25
which give P = 68- 71 x!44 tts., m = 411, a = 1'32,
and observing that the effective vaporization of the
engine has been S = 77 cubic foot of water per
minute, we find, —
1st case. — Effort applied by the engine at
the given velocity, according to our
411 x 0-77 x (6871x144)
theory, — ~298~~ " 10'507 %>*•
Effect produced, including friction and
resistances, as above . » ;.• . 10,103
Difference 404
2nd case. — Effort applied by the engine at
the given velocity, according to our
41 1 x Q- 77 x (68- 71x144)
theory,- 7,215
Effect produced, including friction, &c. 7,346
Difference 131
Mean difference of the two cases . 267
It appears, then, that by this method, the useful
effect is found with a difference only of 267 Ibs., a
very inconsiderable difference in experiments of this
kind, wherein so much depends on the management of
the fire.
2°. To continue the same comparison of the two
theories, let it be required to calculate what quantity
of water per minute the boiler ought to vaporize, to
produce either the first effect or the second. The
method followed by the ordinary theory, again consists
in previously supposing that the volume described by
the piston has been filled with steam at the same
26
pressure as in the boiler, and then in applying to it a
fractional coefficient to account for the losses.
Now, in the first case, the volume described by the
piston at the given velocity, is 1*32 x 298 =393 cubic
feet. Had this volume been filled with steam at the
pressure of the boiler, it would have required a vapor-
393
ization of -~yy ='96 cubic foot of water per minute.
But the real vaporization was but '77 ; wherefore, in
the first case, the coefficient necessary to lead from
the vaporization indicated by the ordinary calculation,
to the real vaporization, 7^- = '81.
In the second case, we find in the same manner, that
the coefficient should be * 55 ; whence, in this problem,
as in the preceding one, no constant coefficient what-
ever can suffice.
Performing, however, the calculation with the mean
coefficient, '68, we find, —
1st case. — Vaporization per minute, calculated by
the ordinary theory, with the coefficient,
1-32x298
— ^p: x-68 -65
Real vaporization '77
Error '12
2nd case. — Vaporization per minute, calculated
by the ordinary theory, with the coefficient,
1-32x434
TTT
Real vaporization , ... . . . . '77
Error '18
27
The mean error committed is then -5- of the vaporiza-
tion, and being, as it is, a mean, it may, in extreme
cases, become f , or amount to half of the whole vapor-
ization.
This is the error committed in seeking a coefficient
expressly for the vaporization. But when the coeffi-
cient, determined in the preceding case, that is, by
the comparison of the theoretical and practical effects,
is used as a divisor, as by many authors it is, much
greater errors are induced, which we will show by an
example farther on.
In our theory, on the contrary, the vaporization
necessary to set in motion the resistance a R at the
velocity v, is given by the formula
~ a R xv
~^TP~-
We have then, —
1st case. — Vaporization calculated from our
10103x298
^eory, 4nx(68.nxl44-
Real vaporization • 77
Difference . . . . . ... . . . -03
2nd case. — Vaporization calculated from our
7346x434
41 lx (68^71x144) •••••'»
Real vaporization -77
Difference -01
3°. Lastly, in the case of finding the velocity of the
piston, supposing the resistance to be given, any me-
thod similar to the ordinary one must inevitably lead
28 .
to errors ; but we must dispense with comparison,
since this problem has never been resolved, and we
shall therefore in this case merely show the verification
of our own theory. The formula relative to this pro-
blem is
= mSP
We find then, —
1st case. — Velocity of the piston in feet per
minute, calculated from our theory,
41 lx -77 x (68-71 x!44)
10103
Real velocity 298
Difference ......... 12
2nd case. — Velocity of the piston from our
411 x-77x(68-71 xl44)
theory, 1jkda * *
Real velocity 434
Difference 8
It consequently appears, that in each of the three
problems in question, our theory leads to the true re-
sult ; whereas the ordinary theory, besides that it leaves
the third problem unresolved, may, in the other two,
lead to very serious errors.
Before abandoning this comparison, we request at-
tention to an effect, in calculating by the ordinary
theory, which we have already mentioned, but which
is here demonstrated, viz., that this calculation gives
the same force applied by the engine in both the cases
considered, notwithstanding their difference of velocity;
29
and such will always be the result, since the calcula-
tion consists merely in multiplying the area of the
piston by the pressure in the boiler, and reducing the
product in a constant proportion. This theory there-
fore maintains, in principle, that the engine can always
draw the same load at all imaginable velocities. Again
we see, that, in the same calculation of the load or
effort applied, the vaporization of the engine does not
appear, which would imply that the engine would
always draw the same load at all velocities, whatever
might be the vaporization of the boiler, which is in-
admissible.
We shall also remark, that in calculating by the
ordinary theory the vaporization of the engine, no
notice is taken of the resistance which the engine is
supposed to move ; so that the vaporization necessary
to draw a given load would be independent of that load
—another result equally impossible.
To these omissions, therefore, or rather to these
errors in principle, are to be attributed the variations
observable in the results given of the ordinary theory
in the examples proposed.
PART II.
ANALYTIC THEORY OF THE STEAM-ENGINE.
ARTICLE I.
CASE OF A GIVEN EXPANSION WITH ANY VELOCITY
OR LOAD WHATEVER.
$ 1 . Of the change of temperature of the steam during
its action in the engine. — When an engine is at work,
the steam is generated in the boiler at a certain
pressure ; it passes from thence into the cylinder,
assuming a different pressure, and, in an expansive
engine, the steam, after its separation from the boiler,
continues to dilate itself more and more in the cylinder,
till the piston is at the end of the stroke. It is
generally supposed, that in all the changes of pressure
which the steam may undergo, its temperature remains
the same ; and it is consequently concluded, that
during the action of the steam in the engine, the
density and volume of that steam follow the law of
Mariotte, namely, that its volume varies in the inverse
31
ratio of the pressure. This supposition greatly sim-
plifies the formulae ; but, as reason and experience
prove it to be altogether inexact, we are compelled to
renounce it, and will substitute in its place another
law, deduced from observation of the facts them-
selves.
We have recognised in a numerous series of ex-
periments, by applying simultaneously a manometer
and a thermometer, both to the boiler of a steam-
engine, and also to the tube through which the steam,
after having terminated its effect, escaped into the
atmosphere, that during all its action in the engine
the steam remains in the state denoted by the name of
saturated steam, that is, at the maximum density for
its temperature. The steam in fact was produced in
the boiler at a very high pressure, and escaped from
the engine at a very low one ; but on its issuing forth,
as well as at the moment of its formation, the ther-
mometer indicated the temperature corresponding to
the pressure marked by the manometer, as if the steam
were immediately generated at the pressure it had at
that moment.
Thus during its whole action in the engine, the
steam remains constantly at the maximum density for
its temperature.
Now, in all steams, the volume depends at once on
the pressure and the temperature ; but in the steam at
the maximum density, the temperature itself depends
on the pressure. It should then be possible to express
the volume of steam of maximum density, in terms of
the pressure alone.
32
The equation which gives the volume of the steam
in any state whatever, in terms of the pressure and
temperature, is very simple : it is deduced from Ma-
riotte's law combined with that of M. Gay-Lussac.
The equation which gives the temperature in terms of
the pressure, for the steam at the maximum density,
is also known : it has heen deduced from the fine
experiments of Messrs. Arago and Dulong on steam
at high pressures, and from those of Southern and
other experimenters on steam produced under low
pressures. By eliminating then the temperature in
these two equations, we shall obtain the analogy re-
quired, which will give immediately, with regard to
steam at the maximum density, for its temperature, the
volume in terms of the pressure alone.
But here arises the difficulty. The equation of the
temperatures is not invariable ; or rather, the same
equation does not apply to all points of the scale.
To be used with accuracy, it requires to be changed
according as the pressure is under that of one atmo-
sphere, or comprised between one and four atmospheres,
or again if it be above four atmospheres. Now when
the steam is acting in an engine, it may happen,
according to the load, or to other conditions of its
motion, that the steam generated at first at a very high
pressure, may act or be expanded in the engine some-
times at a pressure exceeding four atmospheres, some-
times at a pressure less than four atmospheres, but
yet exceeding one, and sometimes at a pressure under
that of one atmosphere. It is impossible then to know
which of the three formulae is to be used in the elimi-
33
nation ; and consequently it is impossible by this
means to attain a general formula representing the
effects of the engine in all cases.
Moreover, were either one of these formulae adopted,
the high radical quantities they contain would so
complicate the calculations as to render them unfit for
practical purposes. And it is to be remarked, that
these divers formulae, after all, are not the expression
of the true mathematical law which connects the tem-
perature and the pressure in saturated steam, but
merely empirical relations, which experiment alone
has demonstrated to have a greater or less degree of
approximation.
A formula of temperatures given by M. Biot is
indeed adapted to all points of the scale, and may be
useful in a great number of delicate researches relative
to the effects of steam ; but as it gives only the
pressure in terms of the temperature, and is, from its
form, incapable of the inverse solution, namely, the
general determination of temperatures in terms of the
pressure, it is unfit for the elimination proposed.
Under these circumstances the only resource is to
seek a direct relation in terms of the pressure alone,
whose results shall represent immediately those of the
two preceding formulas combined ; that is, to calculate
first by means of those formulae a table of volumes of
the steam, and then to seek a direct and simple re-
lation to represent those results. This we have done.
M. Navier had proposed a formula for this purpose.
But that formula, though sufficiently exact in high
pressures, differs widely from experience in pressures
c
34
below that of the atmosphere, which are useful in con-
densing engines ; and it is possible to find one much
more exact for non-condensing engines, namely, that
we are about to offer. We propose then, for this
purpose, the following formulae, in which p represents
the pressure of the steam expressed in pounds per
square foot, and p the ratio of the volume of the steam
to that occupied by the same weight of water :
Formula for high or lo\\n 10000
pressure engines with S M =0-4227 +0-002580
condensation J
Formula for high press- "| 10000
we non-condensing en- [. M = 1.421+0.0023
gmes J
The first formula is equally suitable to pressures
above and below that of the atmosphere, at least within
the limits likely to be considered in applying it to
condensing steam-engines. Those limits are eight or
ten atmospheres for the highest pressures ; and eight
or ten pounds per square inch for the lowest, in con-
sequence of the friction of the engine, the pressure
subsisting against the piston after imperfect con-
densation in the cylinder, and the resistance of the
load. Within these limits then the proposed formula
will be found to give very approximate results.
This first formula might also be applied, without any
error worthy of notice, to non-condensing engines.
But as, in these, the steam can scarcely operate with a
pressure less than two atmospheres, by reason of the
friction of the engine and the resistance of the load, it
is needless to require of the formula exact results of
volumes for pressures under two atmospheres.
35
In this case then the second formula will be found
to give those results with much greater accuracy, and
will consequently be preferred in practice. This will
be readily recognised in a table annexed to the work,
presenting a comparison of the volume of the steam
calculated by the ordinary formulae in terms of the
pressure and temperature, and by the proposed formulae
in terms of the pressure alone.
We state then generally this analogy :
= 1_ , x
n + qp
Consequently, if the steam pass in the engine,
from a certain volume m to another known volume m,
and thereby abandon its primitive pressure F, to as-
sume an unknown pressure p, it is easy to recognise
that the following relation will exist between those
two pressures, and will serve to determine the un-
known quantity p, viz. :
p m' 1 — n fA
"P7" ~7~* 1 -nm' '
This is the relation which we substitute in lieu of
that hitherto employed, and according to which the
volume appears to vary in the inverse ratio of the
pressure. It will be observed that such an hypothesis
may be deduced from the analogy we have just offered,
7/iP
by making n = 0, and q = , m being the volume,
and P the pressure of the steam in the boiler ; for it
is plain that we shall then have,
m P
that is to say, the volumes are inversely as the
pressures.
36
$ 2. Of the divers problems which present themselves
in the calculation of steam-engines. — We distinguish
three cases in an engine : that wherein it works with
a given rate of expansion of the steam, and with a load
or a velocity indefinite ; that in which it works with a
given rate of expansion, and with the load and velocity
proper to produce its maximum of useful effect with
that expansion ; and lastly that wherein, the engine
having been previously regulated for the expansion of
the steam most favourable in that engine, it bears,
moreover, the load most advantageous for that expan-
sion ; which, consequently, produces the absolute
maximum of useful effect in the engine.
We have said that the three fundamental problems
of the calculation of steam-engines consist in finding
successively the velocity, the load, and the vaporiza-
tion of the engine. After the solution of these three
problems, that which first presents itself, as a corollary
to them, consists in determining the useful effect of
the engine, which may be expressed under six different
forms, viz. : by the work done, or the number of
pounds raised one foot high by the engine in a minute ;
by the horse power of the engine ; by the actual duty
or useful effect of one pound of coal ; by the useful
effect of a cubic foot of water converted into steam ;
and by the number of pounds of coal, or of cubic feet
of water, that are necessary to produce one horse
power.
Another research, in fine, no less important, is the
rate of expansion at which the steam must work in an
engine, in order that it may produce given effects.
37
We shall present successively the solution of all these
questions.
The various problems will be resolved in each of
the three cases above mentioned. In the two last, the
question will be to calculate the rate of expansion, the
velocity, the load, and the effects which correspond to
the maximum of, relative or absolute, useful effect of
the engine.
In the ordinary calculations of steam-engines, the
solution of three questions only had been attempted,
viz., — to find the load, the vaporization, and the useful
effect, under its different forms ; which solution is, as
we have seen, faulty. As to the determining of the
velocity for a given load, and that of the rate of expan-
sion for given effects, the calculation of these had not
been proposed. Moreover, the very nature of the
theory employed in those calculations did not allow of
distinguishing, in the machine, the existence of the
three cases which are really found in it. The distinc-
tion we establish may, therefore, at first appear ob-
scure, expressed, as it is, in general terms, and inclu-
ding relations unusual in the consideration of steam-
engines ; but, on a closer view of the question, these
relations will be seen to be of indispensable necessity,
in order to calculate with exactitude either the effects
or the proportions of steam-engines of all systems.
§ 3. Of the velocity of the piston under a given load.
— To embrace at once the most complete mode of
action of the steam, we will suppose an engine work-
ing by expansion, by condensation, and with an in-
definite pressure in the boiler ; and to pass on to
38
unexpansive or uncondensing engines, it will suffice to
make the proper suppressions or substitutions in the
general equations.
From what has been already shown of our theory,
the relations sought between the various data of the
problem are necessarily deduced from two general con-
ditions : the first expressing that the engine has at-
tained a uniform motion, and consequently that the
quantity of labour impressed by the motive power is
equal to the quantity of action developed by the
resistance : the second, that there is a necessary
equality between the emission of steam through the
cylinder and the production by the boiler.
The limits of this extract will not allow us to de-
velop those calculations, simple as they may be ; but
that the proceeding may be understood, we shall state
that, expressing by P the pressure of the steam in the
boiler, and by K the pressure of the same steam in
the cylinder before the expansion, by L the length of
stroke of the piston, and by L' the portion traversed
at the moment the expansion begins, by a the area of
the piston, and by c the clearance of the cylinder, or
the space at each end of the cylinder beyond the portion
traversed by the piston, and which necessarily fills
with steam at each stroke ; lastly, by r the resistance
of the load, by p the pressure subsisting on the other
side of the piston after imperfect condensation, by /
the friction of the engine when not loaded, and by
S the increase of that friction per unit of the load r,
these four forces, as well as the pressures, being more-
over referred to the unit of surface of the piston ; the
39
first of the above conditions produces the following
analogy :
P'«(L' + c) f L' L + c Tl
- - v ' ; ^ :F7 - -flog. =-7-- -- na L >- =
l-»a(L' + c) LL + c L +c J
This equation expressing that the labour developed by
the mover is found entire in the effect produced, be
it remarked, that it is not essentially necessary for the
motion to be strictly uniform. It may equally be
composed of equal oscillations, beginning from no
velocity, and returning to no velocity, provided the
change of velocity take place by insensible degrees, so
as to avoid the loss of vis viva, and that the successive
oscillations be performed in equal times.
As to the second condition of the motion ; if we
denote by S the volume of water vaporized by the
boiler in a unit of time and transmitted to the cylinder,
by m the volume of the steam formed under the press-
ure P of the boiler, compared with the volume of the
same weight of water unvaporized, and by v the velocity
of the piston, the equality between the production of
the steam and its consumption will be found to fur-
nish the second general analogy :
Consequently, by eliminating P' from these two equa
tions, and writing, for greater simplicity,
J-T-. -- naL>
L -t-e
L'
we find definitively :
40
... (1)
an equation which gives the velocity of the motion in
terms of the load and of the other data of the problem.
This formula is quite general, and suits every kind
of steam-engine with continued motion. If the engine
be expansive, L' will be replaced by its value corre-
sponding to the point of the stroke where the steam
begins to be intercepted ; if the engine be unexpansive,
it will suffice to make L'= L, which will give at the
same time K = 1. If it be a condensing engine, p
must stand for the pressure of condensation ; if it be
not a condenser, p will represent the atmospheric press-
ure. And finally, the quantities n and q will have,
according to the case considered, the above-mentioned
value.
§ 4. Of the load and useful effects of the engine. —
If, instead of seeking the velocity in terms of the load
it be required, on the contrary, to know the load suit-
able to a given velocity, the same equation resolved
with reference to r becomes,
L
— nav
3°. To find the vaporization of which the engine
ought to be capable, in order to put in motion a re-
sistance r with a known velocity v, the value of S
must be drawn from the same analogy thus :
S = ?.a
4°. The useful effect produced by the machine, in
41
the unit of time} at the velocity v, is evidently arv.
Hence that useful effect will have for its measure,
5°. If it be desired to know the useful effect, in
horse power, of which the engine is capable at the
velocity v or when loaded with the resistance r, it
suffices to observe that what is called one horse power
represents an effect of 33,000 fts. raised one foot per
minute. All consists then in referring the useful
effect produced by the engine in a unit of time, to the
new unity just chosen, viz. to one horse power ; and
it will consequently suffice to divide the expression
already obtained in the equation (4) by 33,000. Thus,
the useful effect in horse power will be,
uHP. = -0®i _____ (5)
33000
6°. We have just expressed, in the two preceding
questions, the effect of the engine by the work which
it is capable of performing. We are now on the con-
trary about to express that effect by the force which
the engine expends to produce a given quantity of
work. The useful effect of the equation (4) being that
which is due to the volume of water S converted into
steam, in the unit of time, if we suppose that in the
same unit of time N pounds of fuel be consumed, it
is clear that the useful effect produced by each pound
of fuel will be the Nth part of the above effect. It
will then be,
uE. 1 ft. co. = — ....... (6)
42
To apply this formula, it will suffice to know the
quantity of coal consumed in the furnace per minute,
that is, during the production of the vaporization S ;
and this datum may be deduced from a direct experi-
ment on the engine, or from known experiments on
boilers of a similar construction.
7°. The useful effect of the equation (4) being that
which proceeds from the vaporization of the volume
of water S, if it be required to know the useful effect
that will be produced by each cubic foot of water, or
by each unit of S, it will be sufficient to divide the
total effect uE. by the number of units in S. It will
then be,
uE.lft.wa.= ^l ..... (7)
o
8°. In the sixth problem we have obtained the
useful effect produced by one pound of fuel. We
may then, by a simple proportion, deduce from thence
the quantity of fuel necessary to produce one horse
power, viz.
9°. And similarly, the quantity or volume of water
necessary to produce one horse power will be,
Q. wa.for 1 hp. = 330°° S ...... (9)
UJii.
$ 5. Of the expansion of steam, to be adopted in an
expansive engine, in order to produce wanted effects.
10°. Finally, if it be required to know what rate of
expansion the engine must work at, in order to obtain
from it determined effects, the value of L' must be
43
drawn from equation (1). It will be given by the
formula,
v L'+c
-= -- nav — j —
This formula not being of a direct application, we
annex to the work a table which gives its solutions for
the expansion from hundredth to hundredth, with a
very short calculation.
We confine ourselves to these inquiries as being
those which may most commonly be wanted ; but it is
clear that by means of the same general analogies, any
one whatever of the other quantities which figure in
the problem may be determined, as the case may
require. Thus, for instance, may be determined the
area of the piston, or the pressure in the boiler, or the
pressure in the condenser, corresponding to determined
effects of the machine, as has been done for locomo-
tives in our work on that subject.
ART. II.
CASE OF THE MAXIMUM USEFUL EFFECT, WITH
A GIVEN RATE OF EXPANSION.
§ 1. Of the velocity of the maximum -useful effect.
We have resolved the above problems in all their
generality, that is, supposing the engine to move any
load whatever with any velocity whatever, under this
44
single condition, that the load and the velocity be
compatible with the capability of the machine. The
question is now to find what velocity and what load
are most advantageous for the working of the engine,
and what are the effects which, in this case, may be
expected from it ; that is to say, its maxima effects for
a given rate of expansion.
1°. In examining the general expression of the
useful effect produced by the engine at a given velo-
city, we perceive that the expression attains its maxi-
mum for a given rate of expansion when the velocity
is a minimum ; now from the equation (B) the
smallest value of v will be given by P/=P.' The
velocity corresponding to the maximum useful effect
will therefore be,
' * L . (ii)
'
,
Let us however remark, that, mathematically speaking
the pressure P' of the steam in the cylinder can never
be quite equal to P, which is the pressure in the
boiler ; because there exist between the boiler and
the cylinder conduits through which the steam has to
pass, and the passage of these conduits offers a certain
resistance to the motion of the steam ; whence results
that there must exist, on the side of the boiler, a trifling
surplus of pressure equivalent to the overcoming of
the obstacle. But as we have proved elsewhere, that,
with the usual dimensions of engines, this difference
of pressure is not appreciable by the instruments used
to measure the pressure in the boiler, the introduction
of it into the calculations would render the formulae
45
more complicated without making them more exact.
For this reason we neglect that difference here.
The velocity given by the preceding equation is,
then, that at which the engine will produce its maxi-
mum effect for a given expansion. This velocity will
result from the condition P'=P, or reciprocally, when
this velocity takes place in the engine, the steam
enters the cylinder with full pressure, that is, with the
same pressure it has in the boiler. It is necessary to
remark that the velocity of full pressure will not be
the same for all engines ; on the contrary, it will vary
in direct ratio with the vaporization, and in the in-
verse ratio of the area of the cylinder. It may then
occur to be, in one engine, the half or the double of
what it would be in another ; which shows that it is
an error to believe that, because the piston of sta-
tionary engines does not in general exceed a certain
velocity of from 150 to 250 English feet per minute,
the steam of the boiler necessarily reaches the cylinder
with no change of pressure.
It is easy to be seen that a fixed limit, whatever it
may be, cannot in this respect suit all engines ; and
that the only means of knowing the velocity of the
maximum effect, or of full pressure of an engine, is to
calculate it directly for that engine. Such is the
object of the formula we have just given. This for-
mula, moreover, is of a remarkable simplicity, and
requires no other experimental knowledge than that
of the production of steam of which the boiler is
capable.
§ 2. Of the load and maximum useful effect of the
46
engine. — 2°. The useful resistance which the machine is
capable of putting in motion at its velocity of the
maximum effect above, is to be drawn from equation
(2), substituting for v the value just obtained. Calling
the load / we shall find it expressed by
and it is at the same time visible that this load is the
greatest the engine can put in motion with the given
expansion L', for it corresponds to the lowest value of
v in equation (2). Thus, the greatest effect of the
machine, with a given rate of expansion, is attainable
by working the machine at its smallest velocity and
with its maximum load.
It will be observed, that this equation may be used
to determine the friction of the engine without a load,
and its additional friction per unit of the load, upon
the same principles that we have employed in our
Treatise of Locomotive Engines for similar determina-
tions. This is also the mode we propose for steam-
engines of every system.
3°. The vaporization necessary to an engine, in
order to exert a certain maximum effort / at its
minimum velocity v, will be given by equation (3), by
substituting in it r and v, or will be drawn more
simply from equation (11), thus : —
S=(n + qP)m/. L/T+C ---- (13)
Li
4°. The maximum of useful effect producible in the
unit of time, by an engine working with a given ex-
pansion, will be known by formula (4), by introducing
47
for v the velocity proper to produce that effect. Thus
is found,
max.uE. = — - ---- - - I — — (» + /) I ..... (14)
L' + c (1+8) (n + qP) I K U^/'I
It will be observed that this maximum useful effect
depends particularly on the quantity of water S, eva-
porated per minute in the boiler. Hence we see plainly
the error of those who pretend to calculate the useful
effect or the power of engines from the area and the
velocity of the piston, which they set in the place of
the vaporization produced ; this vaporization not only
entering not into their calculation, but forming no part
of their observations.
5°. The useful effect, in horse power, of the engine
will be expressed by
33000
6°. T. 8°. 9°. The various measures of the useful
effect will here be deduced from equations similar to
those (6), (7), (8), and (9).
10°. The expansion at which the engine ought to
be regulated, in order to draw a given load at the
most advantageous velocity, or producing the maxi-
mum of useful effect with that load, will be derived
from equation (12), which gives,
L' + cr L' , L + c-j _ (1+*) r+p+f
~LlL' + c 3'L'+c/~ P
(20)
and the solutions of this formula will be found imme-
diately, and without calculation, by means of the table
given above, as suggested by equation (10).
48
ART. III.
CASE OF THE ABSOLUTE MAXIMUM OF USEFUL
EFFECT.
The preceding inquiries suffice for engines work-
ing without expansion, merely by making L'=L ; be-
cause those engines fall under the case of expansion
fixed a priori. But it is otherwise with engines in
which the rate of expansion may be varied at will.
We have seen that, for a given expansion, the most
advantageous way of working the engine is to give it
the maximum load, which is calculated a priori from
equation (12). Hence we know what load is to be
preferred for every rate of expansion. But the ques-
tion now is to determine, among the various rates of
expansion of which the engine is susceptible, each
accompanied by its corresponding load, which will
produce the greatest useful effect.
For this purpose we must recur to equation (14),
which gives the useful effect produced with a maxi-
mum load /, and seek among all the values assignable
to L', that which will raise the useful effect to a
maximum. Now by making the differential coefficient
of that expression, taken with reference to L', equal
to nothing, we find as the condition of the maximum
sought :
. L + c / L'
— L)
49
This equation will be resolved in the same manner
as the equations (10) and (20), by means of the table
already given ; and after having found the value of
£>
— , it will be introduced in the equations of article
JL
n. ; and the corresponding velocity, load, and useful
effects, will be determined.
However, as the supposition of n=o, q= -- , that
Tfl Jr
is to say, the supposition that the steam preserves its
temperature during its action in the engine, will give
a sufficient approximation in a great many cases, we
present here the corresponding results of all the for-
mulae. They will show, already to a very near degree,
the maximum absolute effects which it is possible to
obtain from an engine, in adopting simultaneously the
most advantageous rate of expansion and the most
advantageous load.
" -— m S _ L P __ Velocity of the absolute maximum
~~' useful effect. '
(22) ar" = a - - Load of the piston corresponding
>T , < ^ to the absolute maximum useful
log __ (L + C> P effect.
LCp+/) + PC
(23) S = ^. MJP+/H-P* Vaporization.
m L P
(24) ab.max.u.E = ayJ'v» —
1 + 8 Absolute maximum of useful ef-
fect'
U-HP — ab' max' u' E Absolute maximum of useful force
33000 in horse power.
/->ru T i — k (/*+./) ^ate °^ expansion which produces
~Y~ these effects,
D
50
The four determinations of the useful effects of a
given quantity of fuel or water will be furnished by
equations similar to those (6), (7), (8), and (9).
The only remark we shah1 make on the subject of
these formulse is, that the load suitable to the pro-
ducing of the absolute maximum useful effect is not
the maximum load that may be imposed on the engine.
In effect, from equation (12), we know that the maxi-
mum load for the engine takes place when L' = L,
and not when
Thus the greatest possible load of the engine is that
of the maximum useful effect without expansion ; but
by applying a lighter load, that of equation (22), and
at the same time the expansion of equation (30), a
still greater useful effect will be obtained.
PART III.
APPLICATION OF THE FORMULA TO THE VARIOUS
SYSTEMS OF STEAM-ENGINES.
WE shall not give here the applications to different
systems of steam-engines, which are developed in this
part of the work. We shall confine ourselves to what
concerns Watt's steam-engines, because they are the
most generally employed in the arts.
Watt's rotative double-acting steam-engine. — These
engines being without expansion, the proper formulae
for calculating their effects will be deduced from the
general formulse by making L' = L, which will give also
K = 1 , and by replacing the quantity p by the pressure
of condensation. We see, moreover, that for these en-
gines, the expansion being susceptible of no variation,
since that detent does not exist, the third case, con-
sidered as to engines in general, cannot occur. There
will be then but two circumstances to consider in their
working, viz., the case wherein they operate with their
maximum load, or load of greatest useful effect, and the
case in which they operate with any load juhatever.
The effects therefore of these engines will visibly be
determined by the following equations :
i
i
s
SD >-}
PH
+ +
V
PH
ffi
a?
ro
GO
a
*
^
H ii
0?
w
'
•o
^
QQ f
OQ
+
+
+ +
s
II
H
53
Although these formula? may at first sight appear
complicated, they will nevertheless be found very
simple in the calculation. It is only necessary to
fix attention to refer all the measures to the same
unit, as will be seen in the following example. It
must be remarked also, that as soon as the velocity
and load of the engine are determined, the useful
effect will be known immediately, being their pro-
duce.
To apply, however, these formulae, some previous
observations are necessary.
In good engines of that system the pressure in the
condenser is usually l'5ft. per square inch, but the
pressure in the cylinder itself, and under the piston,
is in general 2*5fts. more, which gives p=4x 144fts.
It has been deduced, moreover, from a great number
of trials made on Watt's engines, that their friction,
when working with a moderate load, varies from
2*5 fts. per square inch of the piston, in engines of
smaller dimensions; to 1*5 ft), in the more powerful
ones ; which includes the friction of the parts of the
machinery and the force necessary for the action of the
feeding and discharging pumps, &c. By moderate
load in these engines is meant about 8 fts. per square
inch of the piston. Now our experiments on loco-
motives, showing the additional friction of an engine
to be ^ of the resistance, give room to think, that the
additional friction caused in the engine by that load
may be about 1 ft. per square inch. The above in-
formation attributes then to Watt's engines, working
unloaded, a friction of from 1'5 ft. to '5 ft. per square
inch, according to their dimensions, which would give
54
1 tb. for engines of a medium size : this information,
agreeing with what we have deduced from our in-
quiries on locomotives, as has been said above, we
shall continue to admit, in this place, respecting the
friction, the data already indicated in this respect,
viz. : —
/= 1 x 144H5S. 8 = '14.
As an application of these formulae, we will submit
to calculation an engine constructed by Watt at the
Albion Mills near London. The following were its
dimensions : —
Diameter of the cylinder, 34 inches, or a= 6*287
square feet ;
Stroke of the piston, 8 feet, or L=8 feet ;
Clearance of the cylinder, £-0 of the stroke, or c='4
foot;
Effective vaporization, '927 cubic foot of water per
minute, or S='927 cubic foot ;
Consumption of coal in the same time, 6*71 ft>s., or
N = 6'71ft>s. ;
Pressure in the boiler, 16'51bs. per square inch, or
P= 16-5 x 144 fts. ;
Mean pressure of condensation, 4fts. per square
inch, or p=4 x 144 ft>s.
And finally, the engine being a condensing one, we
haverc=-4227 and ^ = '000000258.
The engine had been constructed to work at the
velocity of 256 feet per minute, which was considered
its normal velocity ; but when put to trial by Watt
himself, shortly after its construction, it assumed, in
performing its regular work, esteemed 50 horse-
power, the velocity of 286 feet per minute, consuming
55
at the same time the quantity of water and fuel which
we have just reported.
If then we seek the effects it was capable of pro-
ducing at its velocity of maximum effect, and then at
those of 256 and 286 feet per minute, we shall find,
by the formulae already exposed :
Maximum useful effect.
v = 286 256 v'= 214 Velocity of the piston
in feet per minute ;
ar = 5,621 6,850 9,133 Total load of the pis-
ton in Bbs. ;
r = 6-21 7'57 10-09 Load of the piston in
Bbs. per square inch ;
S = '927 *927 '927 Vaporization in cubic
feet of water per
minute ;
U-E = 1,607,610 1,753,600 1,957,180 Useful effect in fbs.
raised to one foot
per minute.
u-HP = 49 53 59 Useful effect in horse
power.
u.Eiib.Co.= 239,585 261,340 291,680 Useful effect of 1ft. of
coal, in fos. raised to
one foot per minute.
u-Eip. e. = 1,734,200 1,891,700 2,111,300 Useful effect due to
the vaporization of
one cubic foot of
water, in fcs. raised
to one foot per mi-
nute.
Qco.forih. -_ -138 -126 -113 Quantity of coal in
fts., producing the
effect of one horse
power.
Q wa. for 1 h. — • Q 1 9 '017 *0 1 6 Quantity of water, in
cubic feet, producing
the effect of one
horse power.
56
Such are the effects that this engine should produce,
and we see, in consequence, that in performing a
labour estimated at fifty horses, it was to be expected
the engine would acquire the velocity which in fact it
did, viz., that of 286 feet per minute.
Let us now see to what results we should have been
led, had we applied the ordinary calculations to the
experiment of Watt, which we have just reported. In
this experiment, the engine vaporizing *927 cubic foot
of water, and exerting the force of fifty horses, as-
sumed a velocity of 286 feet per minute.
We then find that, since the engine had a useful
effect of no more than fifty horses, and that the theo-
retical force, calculated according to that method,
from the area of the cylinder, the effective pressure in
the boiler, and the velocity of the piston, was,
6-287x (16-5-4) x 144x286
33QOQ =98 horses.
It resulted that, to pass from the theoretical effects
to the practical, it was necessary to use the coefficient
•51. Consequently, by following the reasonings of
that theory, the following conclusions were to be
drawn : —
1°. The observed velocity being 286 feet per minute,
the vaporization calculated on the quantity of water,
which reduced to steam at the pressure of the boiler,
might occupy the volume described by the piston, and
afterwards divided, as is done, by the coefficient, to
take the losses into account, would have been :
isVo*6' 287x286 0 QAK ,. f
i_5J_2 = 2*305 cubic feet per minute,
D 1
instead of -927.
57
2°. The engine having vaporized only '927 cubic
foot of water per minute, the velocity calculated on
the volume of steam formed, at the pressure of the
boiler, and afterwards reduced by the coefficient, not
as has been done, since this problem was not resolved,
but as must naturally be concluded from the significa-
tion attributed to that coefficient, could but be
1 530 x- 927
x * 51 = 1 15 per minute, instead of 28o.
3°. The coefficient found by the comparison of the
theoretical effects to the practical being • 51 , the various
frictions, losses, and resistances of the engine would
amount to • 49 of the effective power ; whereas these
frictions, losses, and resistances, consisting merely of
the friction of the engine and the clearance of the
cylinder, could be estimated only as follows : —
Total friction (including the additional friction)
2 ffis. per square inch, or as a fraction of the
effectual pressure, ^2 . . . . £j . * -17
Clearance of the cylinder, ^ of the effective
force, or . . .:,*,;, ^ * . gj * '. *05
•22
Some authors also employ constant coefficients, not
however using the same to determine the vaporization
as to find the useful effect. This manner of calculating
has arisen from those authors having recognised from
experience, that the steam has in the cylinder a less
pressure and density than in the boiler ; but as they
cannot settle a priori what is that pressure in the
cylinder, and that they always seek to deduce it from
11 PUBLISHED BY JOHN WEALE.
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SAMUEL HALL, Esq., C.E.
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2
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CAPTAIN OLIVER, R.N., Commodore of Bombay War Steam Flotilla.
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CAPTAIN AUSTEN, R.N., late Commander of H.M.S.S. Medea.
GEORGE PEACOCK, Esq., late Master of the Medea.
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Some of the new subjects in this Edition consist of the Works of
Messrs. Boulton and Watt.
The Butterley Company.
Messrs. Maudslay, Son, and Field.
Messrs. Seaward.
Robert Napier, Esq., Glasgow.
Messrs. Fairbairn and Murray.
William Morgan, Esq.
Messrs. Hall, Dartford.
Messrs. Hague.
Messrs. Claude, Girdwoord, and Co.
LIST OF PLATES.
Plate.
1 to 20. The same subjects as in the previous Edition, with several Corrections.
NEW PLATES.
10 B. The several Orders of Parallel Motion.
21. Mr. Kingston's Valves, as fitted on board Sea-going Steam-vessels for Blow-(
Injection, and Hand Pump Sea Valves.
22. Boilers of H.M. Steam Vessel of War African.
23. Boilers of H.M. Steam Vessel of War Medea.
24. Morgan's Paddle Wheel— Seaward' s Paddle Wheel.
25. Positions of a Float of a Radiating Paddle Wheel in a Vessel in Motion — Po
tions of a Float of a Vertically Acting Wheel in a Vessel in Motion.
'?!at!; . TREDGOLD-ewi/i«y«/.
A. ^ycloidal Paddle Wheel fitted to the Great Western Steam Vessel, by Messr*.
Maudslay, Field and Co. Position of a Float of a Cycloidal Paddle Wheel.
27. Captain Oliver's, R.N., Five Points from Courses of Sailing a Steam Vessel.
28. Her Majesty's Steam Ship of War sailing at Different Points in the Wind. 4 views.
29. Trial at sailing her Majesty's Steam Ship of War Medea against the Caledonia,
Vanguard, and Asia.
30. Engine of the Red Rover, Side Elevation, and Plan.
31. Longitudinal Section of ditto.
32. Cross Sections ditto, with Paddles.
33. Side Elevation of the Engine of the Pasha of Egypt's Steam Ship Nile.
34. Plan of ditto, with Paddles.
35. Cross Sections ditto, showing Boilers and Furnace.
36. Section at Paddle Wheel ditto.
37: Plan of Engine of her Majesty's Steam Ship of War Phoenix.
38. Side Elevation of ditto.
39. Cross Section of ditto, showing Paddles and Construction of Ship.
40. Engine of the Ruby Steam Vessel (Gravesend Packet). Plan and Elevation.
41. Section of one of the Engines of the Don Juan, Peninsular Company Packet.
42. Boilers of her Majesty's Ships Hermes, Spitfire, and Firefly.
43. Plan of the Engines of the Imperial Russian Steam Ships Jason and Colchis.
44 Section of ditto.
45. Longitudinal Section of ditto.
46. Section at the Shaft, Section abaft Boilers, ditto.
47. Elevation of Mr. Samuel Hall's Patent Condensing Engines.
48. Section of ditto,
49. Elevation of the Engine of her Majesty's Steam Ship Megaera, fitted with
Messrs. Seaward's Engines and Mr. Samuel Hall's Condenser.
50. Section of ditto.
51. Messrs. Hall's, of Dartford, Engines of the William Wilberforce, Hull and
London Packet, fitted with Mr. Samuel Hall's Condensers. Plan.
52. Elevation of ditto.
53. Cross Section of ditto.
54. Longitudinal Section of ditto.
55 A. Messrs. Hall's, of Dartford, Patent Engines, fitted in the Steam Packet Dartford.
55 B. Ditto, Plan. Elevation ditto.
56 A. Cross Section ditto. Longitudinal Section ditto.
56 B. Ditto. Ditto.
57. Messrs. Fairbairn and Murray's Engine of Twenty Horse Power. Plan.
58. Ditto Section.
59. Ditto Front Elevation and Back Elevation.
60. Messrs. Fairbairn and Co.'s Ten Horse Power Engine. Elevation.
61. Plan. Sectional Plan.
62. Sectional Elevation.
63. Cross Sectional Elevation.
64. Elevation of a Locomotive Engine, manufactured by Messrs. Stephenson, of
Newcastle, for the Stanhope and Tyne Railway.
65. Longitudinal Section of ditto.
66. Spring and Balance Safety Valves.
67 A. Cylinder Cover and Connecting Rods.
67 B. Piston and Cylinder.
68. Boiler Seating of 20-horse Engine, at the manufactory of Messrs. Whitworth
and Co., Manchester.
69, Messrs, Hague's Double ActingiCylinder, Slides^Sections, &c. '
Plate. PUBLIC
12. Bridge near Watford. •
13. Bridge over Railroad at Watford.
14. Viaduct over Colne, near Watford.
15. Bridge over Excavation, South of Watford Tunnel.
16. Boxmoor Skew Bridge.
17 and 18. Sections of Primrose-hill and North Church Tunnels.
19. Architectural Front of Further Entrance.
20 to 29. Working Sections of Embankments and Cuttings on the Line near Blisworth.
30 and 31. Undersetting Rock at Blisworth (Large Scale).
32 and 33. Retaining Walls, Details, &c., ditto.
34 and 35. Undersetting ditto.
36 and 37. West End of Blisworth Cutting.
38. Kilsby Tunnel— Elevation of Entrance.
39. Ditto Section, Details, &c.
40 to 47. Ditto, ditto, ditto.
48. Rails, fifty pounds weight.
49. Ditto, sixty-five pounds weight.
50. Mr. Buck's Chairs.
51. Plans of Crossings from one Line to another.
52. Turnrails.
53. First Class Carriages.
GREAT WESTERN RAILWAY.
J. K. BRUNEL, ESQ., C.E.
54 to 56. Brent Viaduct.
57 and 58. Maidenhead Bridge.
SOUTHAMPTON RAILWAY.
G. LOCK, ESQ., C.E.
59. Occupation Bridge over Railway.
60 and 61. Ditto Bridge under ditto.
62. Embankment.
63. Bridge under Railway.
64. Earth and Timber Waggons.
GREENWICH RAILWAY.
COL. LANDMANN, C.E.
65. Neckinger Viaduct.
66. Section of ditto.
67. Spa Road Viaduct.
68. Section of ditto.
69 and 70. Viaducts and Oblique Arches.
CROYDON RAILWAY.
Jos. GIBBS, ESQ. C.E.
71 and 72. New Cross Bridge — Section of Rails, and Continuous Bearing— Embank-
ments, Earth Carriage, and Details.
THAMES AND BRISTOL JUNCTION RAILWAY.
W. HOSKING, ESQ., C.E.
73 and 74. Tunnels, Bridges, Rails, Chairs, Details, Foundations, &c.
GLASGOW AND GAIRNKIRK.
Messrs. MILLER and GRAINGER, Edinburgh, C.E.
75. Transverse Section at Robroyston, Moss, &c.
76. Comparison of Rails of different Railways.
77. Comet, Locomotive Engine, on Newcastle and Carlisle Railway.
7
Plate- PUBLIC WORKS-co»*<;w«/.
I 79' W^ on*™'8 Engine> Harve>' Combe, on Birmingham Railway.
80.' FlatRai'ls.
81. Losh, Wilson, and Bell's Rail.
82. Ditto Hetton Rail.
83. Sidelings.
CANALS.
W. T. CLARK, ESQ., C.E,
84. Section Thames and Medway Canal.
85. Transit Instruments for ditto.
86. Tunnel Entrance and Cross Section ditto.
87. Tide Locks ditto.
88. Gates, &c., ditto.
89. Centering ditto.
90. Passing Place ditto.
91. Perspective View ditto.
92. Harecastle Tunnel.
93. Montgomeryshire Canal. G. Buck, Esq., C.E.
94. Gloucester and Berkeley Canal. T. Telford, Esq., C.E.
95 to 97. Fishmongers' Hall Wharf. Sir J. Rennie, C.E.
98. Dcptford Pier.
99 to 101. High Bridge (cast-iron), Staffordshire.
102. Double-tumingBridge. R. Walker, Esq., C.E.
103 and 104. Road Bridges.
105. Birmingham and Liverpool Canal.
106. Lock with Single Gate ditto.
107. Gates and Valves ditto.
108. Racks and Pinions ditto.
109. Double Valve ditto.
110. Lever Valve, Rochdale Canal.
111. Mersey and Irwell Boats.
112. Grand Trunk Boat.
1 13 and 1 14. Highgate Road. J. Macneill, Esq., C.E.
115. Geese Bridge Valley.
116, 117, and 118. Roads, Culverts, &c.
119. Coke Ovens.
120. Coking Coal.
121 and 122. Blast Furnace.
123. Lift Hammer.
124 to 127. Steel Furnaces.
128. Mine ditto.
129 and 130. Gas Works. William Richardson, Esq., C.E., Dudley.
131. Liverpool Docks.
PORT OF LONDON.
132. Plan of Port and Harbour.
133. Hemisphere, showing the Position of London as the Centre of the World.
134. London Bridge.
135. Plan of St. Katharine Docks.
136. Gates of ditto.
137- Plan of Gates ditto.
138. Anchor for the Collar of Heel Post and Lock Gates ditto.
139. Cast Iron Swivel Bridge, St. Katharine Docks.
140. Plan of London Docks.
141. Front Elevation of Lock Gates, London Docks.
142. Back ditto ditto.
143. Details ditto ditto.
144. Plan of West India Docks.
145. Plan of East India ditto.
146. Entrance of Lock-gate ditto.
147. Plan of Commercial Docks.
8
Plate. PUBLIC WORKS— continued.
148. New Mooring Chain Lighter.
149. Plan of the Moorings, Deptford Reach.
150. Plan of Mooring Chain— 18 Links, 1 Swivel, and 2 Shackles, &c.
151. Mooring Ships.
152. Section of River.
153. Coffer-dams.
THE HARBOUR AND PORT OF LONDON,
SCIENTIFICALLY, COMMERCIALLY, AND HISTORICALLY DESCRIBED,
Containing Accounts of the History, Privileges, Functions, and Government
thereof; of its Extent, Divisions, and Jurisdictions, Municipal and Commercial ; of
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Tidal and other Observations, and every other necessary Information relative thereto,
accompanied by Charts of the Port and its Dependencies, its Shoals and Soundings,
surveyed by Order of the Port of London Improvement Committee 5 Plans of Docks,
Gates, Piers, Swivel Bridges, Methods of Mooring Vessels, £c., as directed by the
Corporation By-Laws, &c., &c., &c.
By JAMES ELMES,
Architect and Civil Engineer, Surveyor of the Port of London.
22 Plates, large Folio, bound, price II. Is.
Second Edition, with Additional Corrections, in 8vo., with a fine Frontispiece of a
Locomotive Engine, price 8s.
ANALYSIS OF RAILWAYS,
Consisting of Reports of RAILWAYS projected in England and Wales; to which
are added, a Table of Distances from the proposed London Terminus to Eight well-
known Places in the Metropolis, with a copious GLOSSARY, and several Useful
Tables.
By FRANCIS WHISHAW, C.E., M.Inst.C.E.
SECTIO-PLANOGRAPHY.
A DESCRIPTION OF MR. MACNEILL'S METHOD OF LAYING DOWN
RAILWAY SECTIONS AND PLANS IN JUXTA-POSITION,
As adopted by the Standing Order Committee of the House of Commons, 1837.
By FRED. W. SIMMS, Civil Engineer.
With folding Plates, in 4to., price 3s.
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